Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 529 square meters.
A)Dimensions: $\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 6 }$
B)Dimensions: $\frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 3 }$
C)Dimensions: $\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 }$
D)Dimensions: $\frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 }$
E)Dimensions: $\frac { 529 \sqrt { 6 } } { 6 } \times \frac { 529 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 }$

Question

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 529 square meters. 
A)Dimensions: $\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 6 }$
B)Dimensions: $\frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 3 }$
C)Dimensions: $\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 }$
D)Dimensions: $\frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 }$
E)Dimensions: $\frac { 529 \sqrt { 6 } } { 6 } \times \frac { 529 \sqrt { 6 } } { 6 } \times \frac { 23 \sqrt { 6 } } { 6 }$

Quizplus · Accepted Answer

Let the side length of the square base be x and the height be h. Then the surface area is $4x^2+2xh=529$. Solving for h gives $h=\frac{529-4x^2}{2x}$. The volume of the rectangular solid is $V=x^2h=x(\frac{529-4x^2}{2x})=\frac{1}{2}(529x-2x^3)$. Taking the derivative with respect to x and setting it equal to 0, we get $264-6x^2=0$ which gives $x=\frac{23\sqrt{6}}{6}$. Plugging this back into the equation for h, we get $h=\frac{23\sqrt{6}}{3}$. Thus, the dimensions of the rectangular solid are $\frac{23\sqrt{6}}{6} 	imes \frac{23\sqrt{6}}{6}	imes \frac{23\sqrt{6}}{3}$, which is choice D.

Determine the Dimensions of a Rectangular Solid (With a Square 2363×2363×2366\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 6 }3236​​×3236​​×6236​​

Determine the Dimensions of a Rectangular Solid (With a Square $\frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 3 } \times \frac { 23 \sqrt { 6 } } { 6 }$